1 00:00:00,090 --> 00:00:04,320 So to start, we just want to introduce this idea of simple probability. 2 00:00:04,320 --> 00:00:10,980 And at its core, the idea here is that the probability of what we'll call for now event A and event 3 00:00:10,980 --> 00:00:17,880 A could be anything we choose, but the probability of event A occurring is equivalent to the number 4 00:00:17,880 --> 00:00:21,300 of matching outcomes divided by the number of total outcomes. 5 00:00:21,330 --> 00:00:26,670 Now, of course, we need to be specific about what we mean by matching outcomes and total outcomes, 6 00:00:26,670 --> 00:00:33,900 but the idea is that matching outcomes are any outcomes that satisfy event A and total outcomes are 7 00:00:33,900 --> 00:00:39,240 all of the possible equally likely outcomes that could occur in the scenario. 8 00:00:39,270 --> 00:00:45,510 So for instance, let's take a very simple example of flipping a coin, and we're looking for the probability 9 00:00:45,510 --> 00:00:47,640 that we get heads when we flip the coin. 10 00:00:47,640 --> 00:00:53,430 So in this very simple scenario, we flip a two sided coin one time. 11 00:00:53,430 --> 00:00:55,950 On one side is heads, on the other side is tails. 12 00:00:55,950 --> 00:00:59,700 And we want to know the probability that we get heads when we flip that coin. 13 00:00:59,880 --> 00:01:05,220 So in this case, event a from our formula is the event that we get heads. 14 00:01:05,220 --> 00:01:11,430 So we could either say event A is heads and we could calculate the probability of event A like this, 15 00:01:11,430 --> 00:01:15,960 or we could also right it this way, the probability of getting. 16 00:01:16,940 --> 00:01:17,540 Heads. 17 00:01:17,540 --> 00:01:23,780 We don't have to write event A specifically, we can just write the probability of getting heads. 18 00:01:23,780 --> 00:01:31,650 So the probability of flipping heads is equal to well, what are the total outcomes in this scenario? 19 00:01:31,670 --> 00:01:34,630 We can either flip heads or we can flip tails. 20 00:01:34,640 --> 00:01:39,920 There's two possibilities two equally likely outcomes from flipping this coin. 21 00:01:39,920 --> 00:01:45,650 So the total number of all possible outcomes in the scenario is two heads or tails. 22 00:01:45,680 --> 00:01:51,440 This idea then of matching outcomes is the number of outcomes that match our scenario here of Event 23 00:01:51,440 --> 00:01:53,000 A, which is heads. 24 00:01:53,000 --> 00:01:55,550 We're interested in flipping heads. 25 00:01:55,550 --> 00:02:00,170 And so only one of the two scenarios matches that event. 26 00:02:00,200 --> 00:02:06,350 If we flip tails, it doesn't match this event of getting heads only getting heads when we flip the 27 00:02:06,350 --> 00:02:08,240 coin matches this event. 28 00:02:08,240 --> 00:02:13,940 So the number of matching outcomes is simply one, which means that the probability of flipping heads 29 00:02:13,940 --> 00:02:17,300 when we flip a coin one time is one half or. 30 00:02:18,820 --> 00:02:21,460 50%, one half is equal to 50%. 31 00:02:21,490 --> 00:02:25,180 There's a 50% chance we get heads when we flip a coin. 32 00:02:25,600 --> 00:02:32,140 What about a different example here where we roll a six sided die one time and we want a two? 33 00:02:32,170 --> 00:02:36,190 What's the probability that we get a two when we roll that die? 34 00:02:36,220 --> 00:02:40,300 We could write this scenario as the probability of getting a two. 35 00:02:40,330 --> 00:02:43,460 So we might write it like this. 36 00:02:43,480 --> 00:02:49,960 The probability of rolling a two is going to be equal to if we're rolling a six sided die one time. 37 00:02:49,990 --> 00:02:55,190 Of course, there are six total possible outcomes, all of which are equally likely. 38 00:02:55,210 --> 00:02:58,730 We could roll a one, a two, three or four of five or six. 39 00:02:58,750 --> 00:03:02,440 There are six different outcomes we could get when we roll that die. 40 00:03:02,470 --> 00:03:06,400 So six equally likely outcomes, six total outcomes. 41 00:03:06,550 --> 00:03:13,980 And the number of those outcomes that match our criteria here of rolling a two is of course just one, 42 00:03:13,990 --> 00:03:19,750 because if we roll a one, a three for five or six, that doesn't match our criteria. 43 00:03:19,780 --> 00:03:22,300 Only rolling a two matches our criteria. 44 00:03:22,300 --> 00:03:27,960 So there's only one possible outcome, one matching outcome to the one that we're looking for. 45 00:03:27,970 --> 00:03:31,710 So the probability of rolling a two is one over six. 46 00:03:31,720 --> 00:03:37,420 We can also do something a little more complicated where we say, what is the probability of rolling 47 00:03:37,420 --> 00:03:40,490 an even number if we roll that six sided die? 48 00:03:40,510 --> 00:03:43,990 Well, we might write that as the probability of rolling. 49 00:03:45,010 --> 00:03:46,240 An even number. 50 00:03:46,510 --> 00:03:52,510 And we already know from this previous example that there are six possible outcomes when we roll that 51 00:03:52,510 --> 00:03:54,920 die one, two, three, four, five and six. 52 00:03:54,940 --> 00:04:01,690 And of those possible outcomes, only the numbers two, four and six are even numbers one, three and 53 00:04:01,690 --> 00:04:03,000 five are odd numbers. 54 00:04:03,010 --> 00:04:10,630 So the only matching outcomes are the outcomes where we roll a to a four or a six, that's three possibilities. 55 00:04:10,840 --> 00:04:16,269 So the probability in this case is three matching outcomes over six total outcomes. 56 00:04:16,269 --> 00:04:22,510 And then whenever we can, we always want to simplify our fraction, three over six is equivalent to 57 00:04:22,510 --> 00:04:23,440 one half. 58 00:04:23,440 --> 00:04:26,440 So we could write this as one half or. 59 00:04:27,330 --> 00:04:28,200 50%. 60 00:04:28,200 --> 00:04:33,290 So there's a 50% chance there's a one in two chance that we roll an even number. 61 00:04:33,300 --> 00:04:36,960 And then just one more example where we roll a die. 62 00:04:36,990 --> 00:04:42,740 What's the probability that we roll a number on that die that is greater than two? 63 00:04:42,750 --> 00:04:48,870 So we could write this a few ways, but let's just say the probability that X is greater than two, 64 00:04:48,900 --> 00:04:53,430 meaning our die roll results in a number that is greater than two. 65 00:04:53,550 --> 00:04:58,950 Well, we know there are six possible outcomes one, two, three, four, five or six when we roll that 66 00:04:58,950 --> 00:04:59,520 die. 67 00:04:59,520 --> 00:05:01,380 But what about the matching outcomes? 68 00:05:01,380 --> 00:05:07,470 Well, the numbers on the die that are greater than two are the numbers three, four, five and six. 69 00:05:07,470 --> 00:05:11,580 The numbers one and two are not greater than two. 70 00:05:11,580 --> 00:05:18,960 Only three, four, five and six are values greater than two, which means there are four matching outcomes. 71 00:05:18,960 --> 00:05:23,640 And so our probability is four and six or two in. 72 00:05:24,550 --> 00:05:25,420 Three. 73 00:05:25,450 --> 00:05:32,080 A two thirds chance, a two and three chance, or an approximately 67% chance of rolling a number greater 74 00:05:32,080 --> 00:05:37,240 than two When we roll that six sided die one time and we're looking for a number greater than two. 75 00:05:37,630 --> 00:05:44,590 So we understand this general idea that probability of event A, the probability of flipping heads or 76 00:05:44,590 --> 00:05:48,040 rolling A to rolling an even number, rolling a number greater than two. 77 00:05:48,070 --> 00:05:55,300 It's just the likelihood that any of these events happens, given the scenario that we're setting up. 78 00:05:55,300 --> 00:06:00,130 So just think about probability as likelihood of that event occurring. 79 00:06:00,130 --> 00:06:06,970 We also want to be aware at this point, though, of this idea of complimentary probability or the complementary 80 00:06:06,970 --> 00:06:13,900 event and the complement of any event is the scenario where that event does not occur. 81 00:06:13,900 --> 00:06:20,470 And we would write it like this If we're looking for the complement of a we might write it as P. 82 00:06:21,360 --> 00:06:24,120 Of a prime and this would indicate the complement. 83 00:06:24,120 --> 00:06:29,610 So here, if we're looking for the complimentary event of flipping heads, we could write it this way. 84 00:06:30,380 --> 00:06:32,540 Heads and then prime. 85 00:06:32,570 --> 00:06:40,160 This means the complement of the heads event or the probability that we don't flip heads when we flip 86 00:06:40,160 --> 00:06:41,330 a coin one time. 87 00:06:41,330 --> 00:06:43,700 Well, there's two ways to calculate this. 88 00:06:43,700 --> 00:06:49,520 We could, of course, think about going back to our original probability formula and calculating it 89 00:06:49,520 --> 00:06:51,350 manually or from scratch. 90 00:06:51,350 --> 00:06:53,600 So the probability that we don't flip heads. 91 00:06:53,600 --> 00:06:59,120 Well, if we flip a coin one time, we know there are two possible outcomes that stays the same. 92 00:06:59,300 --> 00:07:01,940 The possible outcomes are heads or tails. 93 00:07:01,940 --> 00:07:08,420 The only outcome out of heads or tails that meets this event, not heads, is tails. 94 00:07:08,420 --> 00:07:14,270 And so there's one way that we can match this event, not heads, and that's by flipping tails. 95 00:07:14,270 --> 00:07:17,570 And so the probability of knot heads is one half. 96 00:07:17,570 --> 00:07:23,570 And that makes sense because knot heads, of course, is equal to just tails if we're thinking about 97 00:07:23,570 --> 00:07:24,140 a coin. 98 00:07:24,140 --> 00:07:29,540 And so we could really also write this as the probability of flipping tails and that's one half. 99 00:07:29,540 --> 00:07:35,060 Or to take another example here, we're saying the probability of rolling a two when we roll a six sided 100 00:07:35,060 --> 00:07:37,340 die one time is one over six. 101 00:07:37,340 --> 00:07:40,010 So the probability of. 102 00:07:40,890 --> 00:07:46,860 Not rolling it to is going to be, again, six total outcomes because we're rolling a die one time and 103 00:07:46,860 --> 00:07:51,330 we know that there are six possibilities there, but then not getting a two. 104 00:07:51,360 --> 00:07:59,460 Well, the scenarios that match that are rolling a one, three, four, five or six five different possible 105 00:07:59,460 --> 00:08:00,570 matching outcomes. 106 00:08:00,570 --> 00:08:07,260 So this complementary event here of not to the probability of that complimentary event is five over 107 00:08:07,260 --> 00:08:11,340 six and we could go ahead and calculate that for these other two examples here. 108 00:08:11,550 --> 00:08:17,850 But the thing that we want to notice is that in both of our examples so far, the probability of the 109 00:08:17,850 --> 00:08:25,380 original event and its complement, when we add them together, we always get one or 100%. 110 00:08:25,380 --> 00:08:28,350 So the probability of heads was one half. 111 00:08:28,350 --> 00:08:31,530 The probability of not heads was one half. 112 00:08:31,530 --> 00:08:37,530 So when we add those two things together, we get one half plus one half and we get one. 113 00:08:37,530 --> 00:08:40,440 Or we could call that 100%. 114 00:08:40,590 --> 00:08:46,500 In this scenario, we had the event rolling it two and then the complementary event not rolling it two. 115 00:08:46,530 --> 00:08:54,150 If we add the probabilities of the two complementary events together, we get one over six plus five 116 00:08:54,150 --> 00:08:58,410 over six is equal to six over six or just one or. 117 00:08:59,140 --> 00:09:00,460 100%. 118 00:09:00,550 --> 00:09:03,870 And it turns out that this is always going to be the case. 119 00:09:03,880 --> 00:09:11,260 The sum of the probability of an event and the probability of its complement will always be a total 120 00:09:11,260 --> 00:09:13,840 of one or 100%. 121 00:09:13,960 --> 00:09:20,530 And that makes sense, really, because an event and its complement represents the entire universe of 122 00:09:20,530 --> 00:09:21,610 possibilities, right? 123 00:09:21,610 --> 00:09:22,570 When we roll a die. 124 00:09:22,600 --> 00:09:27,940 There's six possible outcomes if we're talking about the probability of rolling a two and then not rolling 125 00:09:27,940 --> 00:09:28,600 a two. 126 00:09:28,630 --> 00:09:33,820 That, of course, is going to represent all the possibilities when we're flipping a coin, the probability 127 00:09:33,820 --> 00:09:39,550 of getting heads or not getting heads, in other words, getting anything other than heads is going 128 00:09:39,550 --> 00:09:42,420 to represent the entire universe of possibilities. 129 00:09:42,430 --> 00:09:47,260 Over here, when we roll a die and we're looking for an even number, the probability of getting an 130 00:09:47,260 --> 00:09:53,460 even number and then any possibility other than an even number is, of course, going to add to 100%. 131 00:09:53,470 --> 00:09:59,860 It's like saying the probability of this one section of possible outcomes plus the probability of all 132 00:09:59,860 --> 00:10:01,420 the other possible outcomes. 133 00:10:01,420 --> 00:10:07,330 That, of course, is going to give us all possible outcomes or 100% of possible outcomes, which is 134 00:10:07,330 --> 00:10:14,740 why we always end up with these 100% figures when we add the probability of the event and its complement, 135 00:10:14,740 --> 00:10:23,380 which of course leads us into this idea that probability itself is always a value between zero. 136 00:10:24,010 --> 00:10:32,140 And one where zero, a probability of zero means that the event will not occur or that there's a 0% 137 00:10:32,140 --> 00:10:38,140 chance that that event will occur all the way up to one which says that the event will definitely occur, 138 00:10:38,140 --> 00:10:40,680 that there's 100% chance that the event will occur. 139 00:10:40,690 --> 00:10:43,450 And so then we have everything in between. 140 00:10:43,450 --> 00:10:51,520 If we think about this as a percentage where this is 0% all the way over to 100%, it almost makes it 141 00:10:51,520 --> 00:10:55,210 easier to imagine a 0% chance that an event occurs. 142 00:10:55,240 --> 00:10:57,790 A 100% chance that an event occurs. 143 00:10:57,790 --> 00:11:03,190 And then everything in between, like on the coin flip, where there's a 50% chance that we get heads, 144 00:11:03,190 --> 00:11:07,980 a 50% chance we get tails that's right in the middle of this probability scale. 145 00:11:07,990 --> 00:11:14,380 So what this leads us to understand then, is that because summing the probability of an event and its 146 00:11:14,380 --> 00:11:21,280 complement is always going to lead us to this 100% figure, we know that we can always find the probability 147 00:11:21,700 --> 00:11:27,760 of the complement as one minus the probability of the event itself. 148 00:11:27,760 --> 00:11:34,510 Or we could write that in reverse as the probability of event A is equal to one minus the probability 149 00:11:34,510 --> 00:11:36,400 of the compliment. 150 00:11:36,400 --> 00:11:37,930 These formulas are equivalent. 151 00:11:37,930 --> 00:11:41,110 They're just reworked a little bit or rewritten in different ways. 152 00:11:41,110 --> 00:11:47,560 In other words, if we know that the probability of rolling a two is one over six and we want to calculate 153 00:11:47,560 --> 00:11:54,850 the probability of not rolling a two, all we have to do is say the probability of not rolling a two 154 00:11:55,180 --> 00:11:59,350 has to be equal to one minus the probability of rolling a two. 155 00:11:59,380 --> 00:12:04,090 So one minus the probability of rolling a two is one over six. 156 00:12:04,090 --> 00:12:07,630 So we get one minus one over six or. 157 00:12:08,620 --> 00:12:14,770 Five over six, five over six has to be the probability of not rolling it too, which is in fact exactly 158 00:12:14,800 --> 00:12:15,850 what we saw here. 159 00:12:15,880 --> 00:12:22,120 Now that we understand this idea of the probability of an event and the probability of its complement, 160 00:12:22,120 --> 00:12:28,990 and the fact that those two probabilities will always sum to 100%, let's circle back to this original 161 00:12:28,990 --> 00:12:35,050 probability formula and talk about this idea of experimental versus expected probability. 162 00:12:35,260 --> 00:12:42,250 So we just talked about the idea of flipping a coin with two sides, heads and tails, and we said that 163 00:12:42,250 --> 00:12:50,200 the probability of flipping heads was going to be equal to a one in two chance one matching outcome 164 00:12:50,200 --> 00:12:56,740 over two total outcomes or a 50% chance of flipping heads on any one coin flip. 165 00:12:56,770 --> 00:13:06,610 This 50% figure that we calculated and this formula here for the probability of any event A is specifically 166 00:13:06,610 --> 00:13:12,250 what we call expected probability, or we also call it theoretical probability. 167 00:13:12,250 --> 00:13:17,340 It's the probability we expect to have under the given scenario. 168 00:13:17,350 --> 00:13:20,140 But think about this problem here. 169 00:13:20,140 --> 00:13:24,670 Let's say that we're flipping a coin and we flip it five times. 170 00:13:24,670 --> 00:13:28,350 So we flip the coin once and we get heads. 171 00:13:28,360 --> 00:13:35,560 We flip the coin a second time, we get heads, we flip the coin, a third time, we get heads again, 172 00:13:35,560 --> 00:13:40,510 we flip the coin of fourth time, we get heads and then we flip the coin a fifth time and it comes up 173 00:13:40,510 --> 00:13:41,590 tails this time. 174 00:13:41,860 --> 00:13:46,230 So out of our five flips, we got heads, heads, heads, heads, tails. 175 00:13:46,240 --> 00:13:54,730 If we look at this little experiment that we ran, what it would imply to us is that if we flip a coin 176 00:13:55,150 --> 00:14:01,720 five times every five times we flip the coin, we're going to get heads four out of those five times. 177 00:14:01,720 --> 00:14:09,130 And so we might look at this experiment and say that there is a four in five chance or in percentage, 178 00:14:09,580 --> 00:14:15,340 an 80% chance and 80% probability that we get heads when we flip a coin. 179 00:14:15,370 --> 00:14:21,520 Now, of course, intuitively we know that's not true, but if we run this experiment, it makes it 180 00:14:21,520 --> 00:14:25,570 look like there's an 80% chance that we get heads when we flip a coin. 181 00:14:25,570 --> 00:14:28,380 So what exactly is going on here? 182 00:14:28,390 --> 00:14:31,510 Well, we call this 80% number here. 183 00:14:31,690 --> 00:14:34,420 We call this experimental probability. 184 00:14:34,420 --> 00:14:42,070 And experimental probability can vary widely because if we run an experiment where we continue to flip 185 00:14:42,070 --> 00:14:47,440 a coin over and over and over and over again, every time we flip the coin, the probability is going 186 00:14:47,440 --> 00:14:48,110 to change. 187 00:14:48,130 --> 00:14:52,000 In other words, we can see the change happen in real time, right? 188 00:14:52,000 --> 00:14:58,420 So if we flipped the coin five times and we got four out of five outcomes as heads, we would say that 189 00:14:58,420 --> 00:15:02,560 it looks like there's an 80% chance that we get heads when we flip the coin. 190 00:15:02,710 --> 00:15:10,360 Well, if on our very next flip, we flip the coin a sixth time and we get heads, there's now a five 191 00:15:10,360 --> 00:15:16,030 out of six chance of five out of six chance that we get heads when we flip a coin. 192 00:15:16,240 --> 00:15:23,980 But if on our sixth flip we get tails, there's now a four out of six chance that we get heads as we're 193 00:15:23,980 --> 00:15:29,410 flipping this coin, this is an approximately 83% chance of heads. 194 00:15:29,410 --> 00:15:31,570 And this is an approximately. 195 00:15:32,260 --> 00:15:38,560 67% chance of heads, which makes sense because if our probability was sitting at 80% after we flipped 196 00:15:38,560 --> 00:15:45,160 a coin five times and got four of them to come up heads, if we flip heads again, then our experimental 197 00:15:45,160 --> 00:15:47,760 probability that we get heads is increasing. 198 00:15:47,770 --> 00:15:50,850 It now looks like we have an 83% chance of getting heads. 199 00:15:50,860 --> 00:15:54,820 If we flip tails, then that probability of getting heads is going to go down. 200 00:15:54,820 --> 00:15:58,300 It drops from 80% chance to 67% chance. 201 00:15:58,300 --> 00:16:05,110 So we can see our experimental probability moving and changing every time we redo this experiment. 202 00:16:05,110 --> 00:16:11,050 So every single time we flip the coin as we continue this experiment, our experimental probability 203 00:16:11,050 --> 00:16:14,740 of getting heads is going to move in one direction or another. 204 00:16:14,740 --> 00:16:22,450 So that experimental probability can vary widely, but the expected probability stays fixed. 205 00:16:22,450 --> 00:16:28,870 The expected probability is given by this formula here of matching outcomes divided by total outcomes. 206 00:16:28,870 --> 00:16:35,350 And it's the kind of probability that we're used to thinking about What this law of large numbers tells 207 00:16:35,350 --> 00:16:42,820 us is that as we run our experiment more and more and more and more times, our experimental probability 208 00:16:42,820 --> 00:16:47,650 is going to get closer and closer and closer to our expected probability. 209 00:16:47,830 --> 00:16:54,820 So to illustrate that, we could sketch out a graph where we have here along the horizontal axis, our 210 00:16:54,820 --> 00:17:01,180 number of coin flips, and here along the vertical axis, the percentage of flips where we flip heads. 211 00:17:01,180 --> 00:17:06,250 If we think about taking this horizontal axis out to more and more and more experiments. 212 00:17:06,250 --> 00:17:08,829 So here we've only flipped the coin ten times. 213 00:17:08,829 --> 00:17:14,680 Here we've flipped it 50 times all the way up to this far side where we've now flipped the coin 200 214 00:17:14,680 --> 00:17:15,339 times. 215 00:17:15,640 --> 00:17:22,810 What we see is that our data should approach this 50% expected probability because of course we know 216 00:17:22,810 --> 00:17:29,650 the probability of getting heads on any particular coin flip should be 50% as long as we have a fair 217 00:17:29,650 --> 00:17:30,190 coin. 218 00:17:30,190 --> 00:17:32,770 So we should have this line here. 219 00:17:32,770 --> 00:17:39,190 But if we were to sketch in some data from our coin flipping experiment, it's possible it might look 220 00:17:39,190 --> 00:17:40,630 something like this. 221 00:17:40,630 --> 00:17:48,430 When we have a very small number of flips, our percentage of heads might be far from this 50% expected 222 00:17:48,430 --> 00:17:52,780 probability figure and in general, with each additional flip. 223 00:17:52,780 --> 00:17:59,470 So maybe this is where we are at ten flips at the 11th flip, we might move further from this 50% line 224 00:17:59,470 --> 00:18:03,490 or closer to it, and we might go farther away from the line for a while. 225 00:18:03,490 --> 00:18:11,650 But the concept is that eventually, as long as I flip the coin enough times, this data will revert 226 00:18:11,650 --> 00:18:12,190 to the mean. 227 00:18:12,220 --> 00:18:15,310 It'll pull closer and closer into this line. 228 00:18:15,310 --> 00:18:21,520 And even though it might wiggle around it, like we see the data here wiggling around this line, eventually 229 00:18:21,520 --> 00:18:27,010 it's going to converge to this 50% line, which is our expected probability. 230 00:18:27,010 --> 00:18:30,520 So this is our expected. 231 00:18:31,610 --> 00:18:32,720 Probability. 232 00:18:32,990 --> 00:18:36,920 And this is our experimental probability. 233 00:18:38,570 --> 00:18:45,470 Which wiggles in both directions up and down, around expected probability, until eventually it converges 234 00:18:45,470 --> 00:18:51,590 to be closer and closer and closer to expected probability and experimental probability will get closer 235 00:18:51,590 --> 00:18:55,640 to expected probability the further we go out along this horizontal axis. 236 00:18:55,640 --> 00:18:58,750 In other words, the more times we flip the coin. 237 00:18:58,760 --> 00:19:01,190 That is the law of large numbers. 238 00:19:01,190 --> 00:19:06,320 It tells us that experimental probability will get closer and closer to expected probability. 239 00:19:06,320 --> 00:19:09,170 As we run the experiment more and more. 240 00:19:09,200 --> 00:19:16,310 The reason we bring this up experimental versus expected probability is to illustrate basically two 241 00:19:16,310 --> 00:19:17,210 main points. 242 00:19:17,210 --> 00:19:23,780 The first point being that the fewer times we run the experiment, the more variability we can expect 243 00:19:23,780 --> 00:19:29,630 and the further away we can be from expected probability without even necessarily realizing it. 244 00:19:29,630 --> 00:19:36,350 And so this is a good lesson for us to learn when we are running experiments and calculating probability, 245 00:19:36,350 --> 00:19:42,350 the idea being that the smaller the sample size we have, the fewer number of times we flip the coin, 246 00:19:42,350 --> 00:19:44,570 the fewer number of times we roll the die. 247 00:19:44,570 --> 00:19:50,690 If our data is such that we can't necessarily calculate this perfect expected probability number, we 248 00:19:50,690 --> 00:19:52,190 can't fill in this formula. 249 00:19:52,190 --> 00:19:57,830 And instead we have to experiment to see what our probability figure is looking like. 250 00:19:57,830 --> 00:20:04,520 Then we need to realize that the chance of getting an inaccurate figure for probability is higher. 251 00:20:04,520 --> 00:20:10,880 When we're taking smaller samples, when we're sampling a smaller number of data points, which means 252 00:20:10,880 --> 00:20:17,420 our conclusion then is that in order to get a more accurate estimate of probability, we want to take 253 00:20:17,420 --> 00:20:19,940 the largest sample that we possibly can. 254 00:20:19,940 --> 00:20:26,300 If we're flipping a coin, we want to flip it as many times as we possibly can to get a better estimate 255 00:20:26,300 --> 00:20:33,140 of actual expected probability instead of flipping it just a small number of times and relying on whatever 256 00:20:33,140 --> 00:20:36,140 our experimental probability figure happens to be. 257 00:20:36,140 --> 00:20:42,920 So again, our our two main points being that we expect highly variable experimental probability when 258 00:20:42,920 --> 00:20:46,190 we have small samples or small data sets. 259 00:20:46,190 --> 00:20:52,010 And as we increase the size of that data set, the law of large numbers tells us that we'll get closer 260 00:20:52,010 --> 00:20:53,540 to expected probability. 261 00:20:53,540 --> 00:20:59,240 And therefore our conclusion is that we always want to use the largest data set that we possibly can 262 00:20:59,240 --> 00:21:05,060 or that's reasonable to use so that we can get the clearest possible picture of expected probability.